LMFDB - Embedded newform 1728.3.q.j.449.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,3,Mod(449,1728)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 5]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1728.449");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1728.q (of order $6$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$47.0845896815$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	8.0.19269881856.9
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 15x^{6} - 2x^{5} + 133x^{4} - 84x^{3} + 276x^{2} + 144x + 144 $ x^8 - 2x^7 + 15x^6 - 2x^5 + 133x^4 - 84x^3 + 276x^2 + 144*x + 144
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			449.2
Root			$-0.331167 - 0.573598i$ of defining polynomial
Character	$\chi$	$=$	1728.449
Dual form			1728.3.q.j.1601.2

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.0440114 - 0.0254100i) q^{5} +(4.52944 + 7.84521i) q^{7} +O(q^{10})$	$q+(-0.0440114 - 0.0254100i) q^{5} +(4.52944 + 7.84521i) q^{7} +(3.29117 - 1.90016i) q^{11} +(-0.216902 + 0.375686i) q^{13} +26.2355i q^{17} -34.2225 q^{19} +(29.9930 + 17.3164i) q^{23} +(-12.4987 - 21.6484i) q^{25} +(14.0316 - 8.10114i) q^{29} +(-17.1675 + 29.7350i) q^{31} -0.460372i q^{35} -29.2761 q^{37} +(-48.7026 - 28.1185i) q^{41} +(3.94539 + 6.83362i) q^{43} +(33.4489 - 19.3117i) q^{47} +(-16.5316 + 28.6335i) q^{49} +50.5273i q^{53} -0.193132 q^{55} +(-8.54743 - 4.93486i) q^{59} +(36.5718 + 63.3442i) q^{61} +(0.0190923 - 0.0110230i) q^{65} +(12.6797 - 21.9618i) q^{67} -97.8262i q^{71} -77.0599 q^{73} +(29.8143 + 17.2133i) q^{77} +(-42.1389 - 72.9868i) q^{79} +(40.6763 - 23.4845i) q^{83} +(0.666644 - 1.15466i) q^{85} +108.587i q^{89} -3.92978 q^{91} +(1.50618 + 0.869593i) q^{95} +(-32.4021 - 56.1221i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{5} + 6 q^{7}+O(q^{10}) $ 8 * q - 6 * q^5 + 6 * q^7	$ 8 q - 6 q^{5} + 6 q^{7} + 36 q^{11} - 14 q^{13} - 4 q^{19} + 102 q^{23} + 10 q^{25} - 114 q^{29} - 50 q^{31} - 120 q^{37} - 264 q^{41} + 28 q^{43} - 150 q^{47} + 94 q^{49} - 244 q^{55} - 108 q^{59} - 14 q^{61} + 198 q^{65} + 20 q^{67} - 76 q^{73} + 66 q^{77} + 26 q^{79} + 246 q^{83} + 224 q^{85} - 108 q^{91} + 456 q^{95} - 236 q^{97}+O(q^{100}) $ 8 * q - 6 * q^5 + 6 * q^7 + 36 * q^11 - 14 * q^13 - 4 * q^19 + 102 * q^23 + 10 * q^25 - 114 * q^29 - 50 * q^31 - 120 * q^37 - 264 * q^41 + 28 * q^43 - 150 * q^47 + 94 * q^49 - 244 * q^55 - 108 * q^59 - 14 * q^61 + 198 * q^65 + 20 * q^67 - 76 * q^73 + 66 * q^77 + 26 * q^79 + 246 * q^83 + 224 * q^85 - 108 * q^91 + 456 * q^95 - 236 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−0.0440114	−	0.0254100i	−0.00880228	−	0.00508200i	0.495592	−	0.868555i	$-0.334951\pi$
$5$	−0.0440114	−	0.0254100i	−0.00880228	−	0.00508200i	−0.504395	+	0.863473i	$0.668284\pi$
$6$		0			0
$7$	4.52944	+	7.84521i	0.647062	+	1.12074i	0.983821	+	0.179154i	$0.0573360\pi$
$7$	4.52944	+	7.84521i	0.647062	+	1.12074i	−0.336759	+	0.941591i	$0.609331\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.29117	−	1.90016i	0.299198	−	0.172742i	−0.342885	−	0.939377i	$-0.611404\pi$
$11$	3.29117	−	1.90016i	0.299198	−	0.172742i	0.642082	+	0.766636i	$0.278071\pi$
$12$		0			0
$13$	−0.216902	+	0.375686i	−0.0166848	+	0.0288989i	−0.874247	−	0.485481i	$-0.838645\pi$
$13$	−0.216902	+	0.375686i	−0.0166848	+	0.0288989i	0.857562	+	0.514380i	$0.171978\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			26.2355i			1.54327i	0.636068	+	0.771633i	$0.280560\pi$
$17$			26.2355i			1.54327i	−0.636068	+	0.771633i	$0.719440\pi$
$18$		0			0
$19$	−34.2225			−1.80118			−0.900592	−	0.434666i	$-0.856867\pi$
$19$	−34.2225			−1.80118			−0.900592	+	0.434666i	$0.856867\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	29.9930	+	17.3164i	1.30404	+	0.752889i	0.981095	−	0.193528i	$-0.0619931\pi$
$23$	29.9930	+	17.3164i	1.30404	+	0.752889i	0.322947	+	0.946417i	$0.395326\pi$
$24$		0			0
$25$	−12.4987	−	21.6484i	−0.499948	−	0.865936i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	14.0316	−	8.10114i	0.483848	−	0.279350i	−0.238171	−	0.971223i	$-0.576548\pi$
$29$	14.0316	−	8.10114i	0.483848	−	0.279350i	0.722019	+	0.691874i	$0.243214\pi$
$30$		0			0
$31$	−17.1675	+	29.7350i	−0.553790	+	0.959193i	0.444206	+	0.895925i	$0.353486\pi$
$31$	−17.1675	+	29.7350i	−0.553790	+	0.959193i	−0.997997	+	0.0632685i	$0.979848\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		−	0.460372i		−	0.0131535i
$36$		0			0
$37$	−29.2761			−0.791247			−0.395623	−	0.918413i	$-0.629471\pi$
$37$	−29.2761			−0.791247			−0.395623	+	0.918413i	$0.629471\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−48.7026	−	28.1185i	−1.18787	−	0.685816i	−0.230047	−	0.973180i	$-0.573888\pi$
$41$	−48.7026	−	28.1185i	−1.18787	−	0.685816i	−0.957822	+	0.287363i	$0.907221\pi$
$42$		0			0
$43$	3.94539	+	6.83362i	0.0917533	+	0.158921i	0.908249	−	0.418430i	$-0.137420\pi$
$43$	3.94539	+	6.83362i	0.0917533	+	0.158921i	−0.816496	+	0.577352i	$0.804086\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	33.4489	−	19.3117i	0.711678	−	0.410887i	−0.100004	−	0.994987i	$-0.531886\pi$
$47$	33.4489	−	19.3117i	0.711678	−	0.410887i	0.811682	+	0.584100i	$0.198552\pi$
$48$		0			0
$49$	−16.5316	+	28.6335i	−0.337379	+	0.584358i
$50$		0			0
$51$		0			0
$52$		0			0
$53$			50.5273i			0.953344i	0.879081	+	0.476672i	$0.158157\pi$
$53$			50.5273i			0.953344i	−0.879081	+	0.476672i	$0.841843\pi$
$54$		0			0
$55$	−0.193132			−0.00351149
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−8.54743	−	4.93486i	−0.144872	−	0.0836417i	0.425812	−	0.904812i	$-0.359988\pi$
$59$	−8.54743	−	4.93486i	−0.144872	−	0.0836417i	−0.570684	+	0.821170i	$0.693322\pi$
$60$		0			0
$61$	36.5718	+	63.3442i	0.599537	+	1.03843i	0.992889	+	0.119041i	$0.0379821\pi$
$61$	36.5718	+	63.3442i	0.599537	+	1.03843i	−0.393352	+	0.919388i	$0.628685\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.0190923	−	0.0110230i	0.000293728	−	0.000169584i
$66$		0			0
$67$	12.6797	−	21.9618i	0.189249	−	0.327789i	−0.755751	−	0.654859i	$-0.772728\pi$
$67$	12.6797	−	21.9618i	0.189249	−	0.327789i	0.945000	+	0.327070i	$0.106061\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	97.8262i		−	1.37783i	−0.724840	−	0.688917i	$-0.758086\pi$
$71$		−	97.8262i		−	1.37783i	0.724840	−	0.688917i	$-0.241914\pi$
$72$		0			0
$73$	−77.0599			−1.05561			−0.527807	−	0.849364i	$-0.676986\pi$
$73$	−77.0599			−1.05561			−0.527807	+	0.849364i	$0.676986\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	29.8143	+	17.2133i	0.387199	+	0.223549i
$78$		0			0
$79$	−42.1389	−	72.9868i	−0.533404	−	0.923883i	−0.999239	−	0.0390112i	$-0.987579\pi$
$79$	−42.1389	−	72.9868i	−0.533404	−	0.923883i	0.465835	−	0.884872i	$-0.345754\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	40.6763	−	23.4845i	0.490076	−	0.282946i	−0.234530	−	0.972109i	$-0.575355\pi$
$83$	40.6763	−	23.4845i	0.490076	−	0.282946i	0.724606	+	0.689163i	$0.242022\pi$
$84$		0			0
$85$	0.666644	−	1.15466i	0.00784287	−	0.0135843i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			108.587i			1.22008i	0.792371	+	0.610039i	$0.208846\pi$
$89$			108.587i			1.22008i	−0.792371	+	0.610039i	$0.791154\pi$
$90$		0			0
$91$	−3.92978			−0.0431844
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.50618	+	0.869593i	0.0158545	+	0.00915361i
$96$		0			0
$97$	−32.4021	−	56.1221i	−0.334043	−	0.578579i	0.649258	−	0.760568i	$-0.275080\pi$
$97$	−32.4021	−	56.1221i	−0.334043	−	0.578579i	−0.983300	+	0.181990i	$0.941746\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−168.478	+	97.2705i	−1.66809	+	0.963075i	−0.699430	+	0.714701i	$0.746563\pi$
$101$	−168.478	+	97.2705i	−1.66809	+	0.963075i	−0.968664	+	0.248373i	$0.920104\pi$
$102$		0			0
$103$	−12.4420	+	21.5502i	−0.120796	+	0.209225i	−0.920082	−	0.391726i	$-0.871878\pi$
$103$	−12.4420	+	21.5502i	−0.120796	+	0.209225i	0.799286	+	0.600951i	$0.205211\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			23.2306i			0.217108i	0.994091	+	0.108554i	$0.0346221\pi$
$107$			23.2306i			0.217108i	−0.994091	+	0.108554i	$0.965378\pi$
$108$		0			0
$109$	−157.077			−1.44108			−0.720538	−	0.693416i	$-0.756105\pi$
$109$	−157.077			−1.44108			−0.720538	+	0.693416i	$0.756105\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	32.5614	+	18.7994i	0.288154	+	0.166366i	0.637109	−	0.770774i	$-0.280130\pi$
$113$	32.5614	+	18.7994i	0.288154	+	0.166366i	−0.348955	+	0.937140i	$0.613463\pi$
$114$		0			0
$115$	−0.880021	−	1.52424i	−0.00765236	−	0.0132543i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−205.823	+	118.832i	−1.72961	+	0.998589i
$120$		0			0
$121$	−53.2788	+	92.2816i	−0.440321	+	0.762658i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			2.54087i			0.0203269i
$126$		0			0
$127$	48.4364			0.381389			0.190694	−	0.981649i	$-0.438926\pi$
$127$	48.4364			0.381389			0.190694	+	0.981649i	$0.438926\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	0.0274376	+	0.0158411i	0.000209447	+	0.000120924i	0.500105	−	0.865965i	$-0.333295\pi$
$131$	0.0274376	+	0.0158411i	0.000209447	+	0.000120924i	−0.499895	+	0.866086i	$0.666628\pi$
$132$		0			0
$133$	−155.009	−	268.483i	−1.16548	−	2.01867i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−0.913705	+	0.527528i	−0.00666938	+	0.00385057i	−0.503331	−	0.864094i	$-0.667892\pi$
$137$	−0.913705	+	0.527528i	−0.00666938	+	0.00385057i	0.496662	+	0.867944i	$0.334559\pi$
$138$		0			0
$139$	−45.3655	+	78.5754i	−0.326371	+	0.565290i	−0.981789	−	0.189976i	$-0.939159\pi$
$139$	−45.3655	+	78.5754i	−0.326371	+	0.565290i	0.655418	+	0.755266i	$0.272492\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$			1.64860i			0.0115286i
$144$		0			0
$145$	−0.823399			−0.00567862
$146$		0			0
$147$		0			0
$148$		0			0
$149$	15.1086	+	8.72295i	0.101400	+	0.0585433i	0.549842	−	0.835268i	$-0.314688\pi$
$149$	15.1086	+	8.72295i	0.101400	+	0.0585433i	−0.448442	+	0.893812i	$0.648021\pi$
$150$		0			0
$151$	40.8713	+	70.7912i	0.270671	+	0.468816i	0.969034	−	0.246928i	$-0.0794211\pi$
$151$	40.8713	+	70.7912i	0.270671	+	0.468816i	−0.698363	+	0.715744i	$0.746088\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	1.51113	−	0.872452i	0.00974923	−	0.00562872i
$156$		0			0
$157$	−96.4835	+	167.114i	−0.614544	+	1.06442i	0.375920	+	0.926652i	$0.377327\pi$
$157$	−96.4835	+	167.114i	−0.614544	+	1.06442i	−0.990464	+	0.137770i	$0.956007\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			313.735i			1.94866i
$162$		0			0
$163$	165.401			1.01473			0.507364	−	0.861732i	$-0.330620\pi$
$163$	165.401			1.01473			0.507364	+	0.861732i	$0.330620\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	215.643	+	124.502i	1.29128	+	0.745520i	0.978881	−	0.204432i	$-0.0655346\pi$
$167$	215.643	+	124.502i	1.29128	+	0.745520i	0.312398	+	0.949951i	$0.398868\pi$
$168$		0			0
$169$	84.4059	+	146.195i	0.499443	+	0.865061i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−117.476	+	67.8248i	−0.679052	+	0.392051i	−0.799498	−	0.600669i	$-0.794901\pi$
$173$	−117.476	+	67.8248i	−0.679052	+	0.392051i	0.120446	+	0.992720i	$0.461568\pi$
$174$		0			0
$175$	113.224	−	196.110i	0.646995	−	1.12063i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	95.4526i		−	0.533255i	−0.963800	−	0.266627i	$-0.914091\pi$
$179$		−	95.4526i		−	0.533255i	0.963800	−	0.266627i	$-0.0859093\pi$
$180$		0			0
$181$	−58.9249			−0.325552			−0.162776	−	0.986663i	$-0.552045\pi$
$181$	−58.9249			−0.325552			−0.162776	+	0.986663i	$0.552045\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.28848	+	0.743906i	0.00696477	+	0.00402111i
$186$		0			0
$187$	49.8517	+	86.3457i	0.266587	+	0.461742i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−164.852	+	95.1775i	−0.863101	+	0.498311i	−0.865049	−	0.501687i	$-0.832713\pi$
$191$	−164.852	+	95.1775i	−0.863101	+	0.498311i	0.00194880	+	0.999998i	$0.499380\pi$
$192$		0			0
$193$	5.29645	−	9.17373i	0.0274428	−	0.0475323i	−0.851978	−	0.523578i	$-0.824597\pi$
$193$	5.29645	−	9.17373i	0.0274428	−	0.0475323i	0.879421	+	0.476046i	$0.157930\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			215.874i			1.09581i	0.836541	+	0.547904i	$0.184574\pi$
$197$			215.874i			1.09581i	−0.836541	+	0.547904i	$0.815426\pi$
$198$		0			0
$199$	−146.668			−0.737026			−0.368513	−	0.929623i	$-0.620133\pi$
$199$	−146.668			−0.737026			−0.368513	+	0.929623i	$0.620133\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	127.110	+	73.3872i	0.626159	+	0.361513i
$204$		0			0
$205$	1.42898	+	2.47506i	0.00697063	+	0.0120735i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−112.632	+	65.0282i	−0.538910	+	0.311140i
$210$		0			0
$211$	54.8335	−	94.9744i	0.259874	−	0.450116i	−0.706334	−	0.707879i	$-0.749652\pi$
$211$	54.8335	−	94.9744i	0.259874	−	0.450116i	0.966208	+	0.257763i	$0.0829855\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	0.401009i		−	0.00186516i
$216$		0			0
$217$	−311.036			−1.43335
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.85631	−	5.69054i	−0.0445987	−	0.0257491i
$222$		0			0
$223$	73.8403	+	127.895i	0.331123	+	0.573521i	0.982732	−	0.185033i	$-0.0592393\pi$
$223$	73.8403	+	127.895i	0.331123	+	0.573521i	−0.651610	+	0.758554i	$0.725906\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−346.255	+	199.911i	−1.52535	+	0.880664i	−0.525806	+	0.850605i	$0.676236\pi$
$227$	−346.255	+	199.911i	−1.52535	+	0.880664i	−0.999548	+	0.0300589i	$0.990431\pi$
$228$		0			0
$229$	−39.1692	+	67.8430i	−0.171044	+	0.296258i	−0.938785	−	0.344503i	$-0.888047\pi$
$229$	−39.1692	+	67.8430i	−0.171044	+	0.296258i	0.767741	+	0.640760i	$0.221381\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	352.995i		−	1.51500i	−0.652835	−	0.757500i	$-0.726420\pi$
$233$		−	352.995i		−	1.51500i	0.652835	−	0.757500i	$-0.273580\pi$
$234$		0			0
$235$	−1.96284			−0.00835251
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−279.549	−	161.397i	−1.16966	−	0.675303i	−0.216060	−	0.976380i	$-0.569321\pi$
$239$	−279.549	−	161.397i	−1.16966	−	0.675303i	−0.953600	+	0.301077i	$0.902654\pi$
$240$		0			0
$241$	−105.601	−	182.907i	−0.438180	−	0.758949i	0.559370	−	0.828918i	$-0.311043\pi$
$241$	−105.601	−	182.907i	−0.438180	−	0.758949i	−0.997549	+	0.0699691i	$0.977710\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	1.45516	−	0.840135i	0.00593941	−	0.00342912i
$246$		0			0
$247$	7.42293	−	12.8569i	0.0300524	−	0.0520522i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			206.824i			0.824001i	0.911184	+	0.412001i	$0.135170\pi$
$251$			206.824i			0.824001i	−0.911184	+	0.412001i	$0.864830\pi$
$252$		0			0
$253$	131.616			0.520222
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−20.7432	−	11.9761i	−0.0807127	−	0.0465995i	0.459100	−	0.888384i	$-0.348172\pi$
$257$	−20.7432	−	11.9761i	−0.0807127	−	0.0465995i	−0.539813	+	0.841785i	$0.681505\pi$
$258$		0			0
$259$	−132.604	−	229.678i	−0.511986	−	0.886786i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	119.369	−	68.9175i	0.453873	−	0.262044i	−0.255592	−	0.966785i	$-0.582270\pi$
$263$	119.369	−	68.9175i	0.453873	−	0.262044i	0.709464	+	0.704741i	$0.248937\pi$
$264$		0			0
$265$	1.28390	−	2.22377i	0.00484489	−	0.00839160i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	249.461i		−	0.927366i	−0.886001	−	0.463683i	$-0.846528\pi$
$269$		−	249.461i		−	0.927366i	0.886001	−	0.463683i	$-0.153472\pi$
$270$		0			0
$271$	72.6700			0.268155			0.134078	−	0.990971i	$-0.457193\pi$
$271$	72.6700			0.268155			0.134078	+	0.990971i	$0.457193\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−82.2709	−	47.4991i	−0.299167	−	0.172724i
$276$		0			0
$277$	182.021	+	315.270i	0.657117	+	1.13816i	0.981359	+	0.192185i	$0.0615574\pi$
$277$	182.021	+	315.270i	0.657117	+	1.13816i	−0.324242	+	0.945974i	$0.605109\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−241.120	+	139.211i	−0.858077	+	0.495411i	−0.863368	−	0.504575i	$-0.831649\pi$
$281$	−241.120	+	139.211i	−0.858077	+	0.495411i	0.00529060	+	0.999986i	$0.498316\pi$
$282$		0			0
$283$	13.7745	−	23.8581i	0.0486732	−	0.0843044i	−0.840662	−	0.541560i	$-0.817834\pi$
$283$	13.7745	−	23.8581i	0.0486732	−	0.0843044i	0.889336	+	0.457255i	$0.151167\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	509.443i		−	1.77506i
$288$		0			0
$289$	−399.303			−1.38167
$290$		0			0
$291$		0			0
$292$		0			0
$293$	342.145	+	197.537i	1.16773	+	0.674189i	0.953144	−	0.302517i	$-0.0978267\pi$
$293$	342.145	+	197.537i	1.16773	+	0.674189i	0.214585	+	0.976705i	$0.431160\pi$
$294$		0			0
$295$	0.250789	+	0.434380i	0.000850133	+	0.00147247i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−13.0111	+	7.51195i	−0.0435153	+	0.0251236i
$300$		0			0
$301$	−35.7408	+	61.9049i	−0.118740	+	0.205664i
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	3.71715i		−	0.0121874i
$306$		0			0
$307$	122.443			0.398836			0.199418	−	0.979915i	$-0.436095\pi$
$307$	122.443			0.398836			0.199418	+	0.979915i	$0.436095\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	420.591	+	242.829i	1.35238	+	0.780799i	0.988583	−	0.150677i	$-0.0481455\pi$
$311$	420.591	+	242.829i	1.35238	+	0.780799i	0.363801	+	0.931477i	$0.381479\pi$
$312$		0			0
$313$	5.15434	+	8.92759i	0.0164676	+	0.0285226i	0.874142	−	0.485671i	$-0.161425\pi$
$313$	5.15434	+	8.92759i	0.0164676	+	0.0285226i	−0.857674	+	0.514194i	$0.828091\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−144.879	+	83.6462i	−0.457033	+	0.263868i	−0.710796	−	0.703398i	$-0.751665\pi$
$317$	−144.879	+	83.6462i	−0.457033	+	0.263868i	0.253763	+	0.967266i	$0.418332\pi$
$318$		0			0
$319$	30.7869	−	53.3245i	0.0965107	−	0.167162i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		−	897.845i		−	2.77971i
$324$		0			0
$325$	10.8440			0.0333661
$326$		0			0
$327$		0			0
$328$		0			0
$329$	303.009	+	174.942i	0.921000	+	0.531740i
$330$		0			0
$331$	−52.2422	−	90.4861i	−0.157831	−	0.273372i	0.776255	−	0.630419i	$-0.217117\pi$
$331$	−52.2422	−	90.4861i	−0.157831	−	0.273372i	−0.934086	+	0.357047i	$0.883784\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−1.11610	+	0.644381i	−0.00333164	+	0.00192352i
$336$		0			0
$337$	196.086	−	339.631i	0.581858	−	1.00781i	−0.413401	−	0.910549i	$-0.635659\pi$
$337$	196.086	−	339.631i	0.581858	−	1.00781i	0.995259	−	0.0972586i	$-0.0310074\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			130.484i			0.382651i
$342$		0			0
$343$	144.370			0.420903
$344$		0			0
$345$		0			0
$346$		0			0
$347$	247.220	+	142.732i	0.712449	+	0.411333i	0.811967	−	0.583703i	$-0.198397\pi$
$347$	247.220	+	142.732i	0.712449	+	0.411333i	−0.0995180	+	0.995036i	$0.531730\pi$
$348$		0			0
$349$	−151.562	−	262.514i	−0.434276	−	0.752189i	0.562960	−	0.826484i	$-0.309663\pi$
$349$	−151.562	−	262.514i	−0.434276	−	0.752189i	−0.997236	+	0.0742956i	$0.976329\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−308.183	+	177.929i	−0.873039	+	0.504049i	−0.868357	−	0.495940i	$-0.834824\pi$
$353$	−308.183	+	177.929i	−0.873039	+	0.504049i	−0.00468222	+	0.999989i	$0.501490\pi$
$354$		0			0
$355$	−2.48576	+	4.30547i	−0.00700215	+	0.0121281i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			148.171i			0.412733i	0.978475	+	0.206367i	$0.0661639\pi$
$359$			148.171i			0.412733i	−0.978475	+	0.206367i	$0.933836\pi$
$360$		0			0
$361$	810.179			2.24426
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.39151	+	1.95809i	0.00929181	+	0.00536463i
$366$		0			0
$367$	−123.772	−	214.380i	−0.337254	−	0.584141i	0.646661	−	0.762777i	$-0.276165\pi$
$367$	−123.772	−	214.380i	−0.337254	−	0.584141i	−0.983915	+	0.178636i	$0.942831\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−396.397	+	228.860i	−1.06846	+	0.616873i
$372$		0			0
$373$	224.520	−	388.881i	0.601931	−	1.04258i	−0.390597	−	0.920562i	$-0.627731\pi$
$373$	224.520	−	388.881i	0.601931	−	1.04258i	0.992528	−	0.122014i	$-0.0389353\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$			7.02862i			0.0186436i
$378$		0			0
$379$	−618.282			−1.63135			−0.815675	−	0.578510i	$-0.803634\pi$
$379$	−618.282			−1.63135			−0.815675	+	0.578510i	$0.803634\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	370.803	+	214.083i	0.968155	+	0.558965i	0.898673	−	0.438619i	$-0.144532\pi$
$383$	370.803	+	214.083i	0.968155	+	0.558965i	0.0694818	+	0.997583i	$0.477865\pi$
$384$		0			0
$385$	−0.874780	−	1.51516i	−0.00227216	−	0.00393549i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	585.313	−	337.930i	1.50466	−	0.868716i	0.504674	−	0.863310i	$-0.331613\pi$
$389$	585.313	−	337.930i	1.50466	−	0.868716i	0.999985	−	0.00540555i	$-0.00172065\pi$
$390$		0			0
$391$	−454.306	+	786.881i	−1.16191	+	2.01248i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			4.28300i			0.0108430i
$396$		0			0
$397$	−209.902			−0.528721			−0.264361	−	0.964424i	$-0.585161\pi$
$397$	−209.902			−0.528721			−0.264361	+	0.964424i	$0.585161\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−175.023	−	101.050i	−0.436467	−	0.251994i	0.265631	−	0.964075i	$-0.414420\pi$
$401$	−175.023	−	101.050i	−0.436467	−	0.251994i	−0.702098	+	0.712080i	$0.747753\pi$
$402$		0			0
$403$	−7.44734	−	12.8992i	−0.0184797	−	0.0320079i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−96.3528	+	55.6293i	−0.236739	+	0.136681i
$408$		0			0
$409$	−291.252	+	504.464i	−0.712108	+	1.23341i	0.251957	+	0.967739i	$0.418926\pi$
$409$	−291.252	+	504.464i	−0.712108	+	1.23341i	−0.964064	+	0.265669i	$0.914407\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		−	89.4085i		−	0.216486i
$414$		0			0
$415$	−2.38696			−0.00575172
$416$		0			0
$417$		0			0
$418$		0			0
$419$	675.460	+	389.977i	1.61208	+	0.930733i	0.988888	+	0.148662i	$0.0474968\pi$
$419$	675.460	+	389.977i	1.61208	+	0.930733i	0.623189	+	0.782071i	$0.285837\pi$
$420$		0			0
$421$	84.1068	+	145.677i	0.199779	+	0.346027i	0.948457	−	0.316907i	$-0.102644\pi$
$421$	84.1068	+	145.677i	0.199779	+	0.346027i	−0.748678	+	0.662934i	$0.769311\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	567.957	−	327.910i	1.33637	−	0.771553i
$426$		0			0
$427$	−331.299	+	573.827i	−0.775876	+	1.34386i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			182.732i			0.423973i	0.977273	+	0.211986i	$0.0679933\pi$
$431$			182.732i			0.423973i	−0.977273	+	0.211986i	$0.932007\pi$
$432$		0			0
$433$	447.193			1.03278			0.516389	−	0.856354i	$-0.327276\pi$
$433$	447.193			1.03278			0.516389	+	0.856354i	$0.327276\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1026.43	−	592.612i	−2.34882	−	1.35609i
$438$		0			0
$439$	98.6108	+	170.799i	0.224626	+	0.389063i	0.956207	−	0.292691i	$-0.0945507\pi$
$439$	98.6108	+	170.799i	0.224626	+	0.389063i	−0.731581	+	0.681754i	$0.761217\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	244.803	−	141.337i	0.552603	−	0.319045i	−0.197568	−	0.980289i	$-0.563304\pi$
$443$	244.803	−	141.337i	0.552603	−	0.319045i	0.750171	+	0.661244i	$0.229971\pi$
$444$		0			0
$445$	2.75919	−	4.77906i	0.00620043	−	0.0107395i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	349.046i		−	0.777386i	−0.921367	−	0.388693i	$-0.872927\pi$
$449$		−	349.046i		−	0.777386i	0.921367	−	0.388693i	$-0.127073\pi$
$450$		0			0
$451$	−213.718			−0.473877
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.172955	+	0.0998556i	0.000380121	+	0.000219463i
$456$		0			0
$457$	−40.2987	−	69.7995i	−0.0881811	−	0.152734i	0.818561	−	0.574419i	$-0.194772\pi$
$457$	−40.2987	−	69.7995i	−0.0881811	−	0.152734i	−0.906742	+	0.421685i	$0.861439\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	495.135	−	285.866i	1.07405	−	0.620100i	0.144761	−	0.989467i	$-0.453759\pi$
$461$	495.135	−	285.866i	1.07405	−	0.620100i	0.929284	+	0.369366i	$0.120425\pi$
$462$		0			0
$463$	139.837	−	242.205i	0.302024	−	0.523120i	−0.674571	−	0.738210i	$-0.735671\pi$
$463$	139.837	−	242.205i	0.302024	−	0.523120i	0.976594	+	0.215090i	$0.0690045\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$		−	506.702i		−	1.08501i	−0.840051	−	0.542507i	$-0.817475\pi$
$467$		−	506.702i		−	1.08501i	0.840051	−	0.542507i	$-0.182525\pi$
$468$		0			0
$469$	229.727			0.489823
$470$		0			0
$471$		0			0
$472$		0			0
$473$	25.9699	+	14.9938i	0.0549048	+	0.0316993i
$474$		0			0
$475$	427.737	+	740.862i	0.900499	+	1.55971i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	217.596	−	125.629i	0.454271	−	0.262274i	−0.255361	−	0.966846i	$-0.582194\pi$
$479$	217.596	−	125.629i	0.454271	−	0.262274i	0.709632	+	0.704572i	$0.248861\pi$
$480$		0			0
$481$	6.35006	−	10.9986i	0.0132018	−	0.0228662i
$482$		0			0
$483$		0			0
$484$		0			0
$485$			3.29335i			0.00679041i
$486$		0			0
$487$	−313.224			−0.643170			−0.321585	−	0.946881i	$-0.604216\pi$
$487$	−313.224			−0.643170			−0.321585	+	0.946881i	$0.604216\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	533.397	+	307.957i	1.08635	+	0.627204i	0.932602	−	0.360906i	$-0.117533\pi$
$491$	533.397	+	307.957i	1.08635	+	0.627204i	0.153747	+	0.988110i	$0.450866\pi$
$492$		0			0
$493$	212.538	+	368.126i	0.431111	+	0.746706i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	767.467	−	443.098i	1.54420	−	0.891544i
$498$		0			0
$499$	412.029	−	713.654i	0.825709	−	1.43017i	−0.0756679	−	0.997133i	$-0.524109\pi$
$499$	412.029	−	713.654i	0.825709	−	1.43017i	0.901377	−	0.433036i	$-0.142558\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	175.718i		−	0.349340i	−0.984627	−	0.174670i	$-0.944114\pi$
$503$		−	175.718i		−	0.349340i	0.984627	−	0.174670i	$-0.0558858\pi$
$504$		0			0
$505$	9.88657			0.0195774
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−314.934	−	181.827i	−0.618730	−	0.357224i	0.157644	−	0.987496i	$-0.449610\pi$
$509$	−314.934	−	181.827i	−0.618730	−	0.357224i	−0.776374	+	0.630272i	$0.782943\pi$
$510$		0			0
$511$	−349.038	−	604.551i	−0.683049	−	1.18307i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.09518	−	0.632301i	0.00212656	−	0.00122777i
$516$		0			0
$517$	73.3907	−	127.116i	0.141955	−	0.245873i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	458.709i		−	0.880440i	−0.897890	−	0.440220i	$-0.854901\pi$
$521$		−	458.709i		−	0.880440i	0.897890	−	0.440220i	$-0.145099\pi$
$522$		0			0
$523$	458.289			0.876270			0.438135	−	0.898909i	$-0.355639\pi$
$523$	458.289			0.876270			0.438135	+	0.898909i	$0.355639\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−780.113	−	450.398i	−1.48029	−	0.854646i
$528$		0			0
$529$	335.219	+	580.616i	0.633683	+	1.09757i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	21.1274	−	12.1979i	0.0396387	−	0.0228854i
$534$		0			0
$535$	0.590289	−	1.02241i	0.00110334	−	0.00191105i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			125.651i			0.233118i
$540$		0			0
$541$	824.876			1.52472			0.762362	−	0.647150i	$-0.224039\pi$
$541$	824.876			1.52472			0.762362	+	0.647150i	$0.224039\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	6.91319	+	3.99133i	0.0126847	+	0.00732354i
$546$		0			0
$547$	16.8719	+	29.2230i	0.0308444	+	0.0534241i	0.881036	−	0.473050i	$-0.156847\pi$
$547$	16.8719	+	29.2230i	0.0308444	+	0.0534241i	−0.850191	+	0.526474i	$0.823514\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−480.196	+	277.241i	−0.871499	+	0.503160i
$552$		0			0
$553$	381.731	−	661.178i	0.690291	−	1.19562i
$554$		0			0
$555$		0			0
$556$		0			0
$557$			541.032i			0.971332i	0.874145	+	0.485666i	$0.161423\pi$
$557$			541.032i			0.971332i	−0.874145	+	0.485666i	$0.838577\pi$
$558$		0			0
$559$	−3.42306			−0.00612354
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−97.8909	−	56.5173i	−0.173874	−	0.100386i	0.410537	−	0.911844i	$-0.365341\pi$
$563$	−97.8909	−	56.5173i	−0.173874	−	0.100386i	−0.584411	+	0.811458i	$0.698674\pi$
$564$		0			0
$565$	−0.955383	−	1.65477i	−0.00169094	−	0.00292880i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−236.524	+	136.557i	−0.415684	+	0.239995i	−0.693229	−	0.720717i	$-0.743813\pi$
$569$	−236.524	+	136.557i	−0.415684	+	0.239995i	0.277545	+	0.960713i	$0.410479\pi$
$570$		0			0
$571$	−122.654	+	212.443i	−0.214806	+	0.372054i	−0.953212	−	0.302301i	$-0.902245\pi$
$571$	−122.654	+	212.443i	−0.214806	+	0.372054i	0.738407	+	0.674356i	$0.235579\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$		−	865.733i		−	1.50562i
$576$		0			0
$577$	632.666			1.09648			0.548238	−	0.836323i	$-0.315299\pi$
$577$	632.666			1.09648			0.548238	+	0.836323i	$0.315299\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	368.482	+	212.743i	0.634220	+	0.366167i
$582$		0			0
$583$	96.0099	+	166.294i	0.164682	+	0.285238i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−980.129	+	565.878i	−1.66973	+	0.964017i	−0.701942	+	0.712234i	$0.747683\pi$
$587$	−980.129	+	565.878i	−1.66973	+	0.964017i	−0.967784	+	0.251782i	$0.918983\pi$
$588$		0			0
$589$	587.515	−	1017.61i	0.997478	−	1.72768i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	180.213i		−	0.303900i	−0.988388	−	0.151950i	$-0.951445\pi$
$593$		−	180.213i		−	0.303900i	0.988388	−	0.151950i	$-0.0485553\pi$
$594$		0			0
$595$	12.0781			0.0202993
$596$		0			0
$597$		0			0
$598$		0			0
$599$	566.086	+	326.830i	0.945052	+	0.545626i	0.891541	−	0.452941i	$-0.149625\pi$
$599$	566.086	+	326.830i	0.945052	+	0.545626i	0.0535119	+	0.998567i	$0.482959\pi$
$600$		0			0
$601$	−178.947	−	309.945i	−0.297749	−	0.515716i	0.677872	−	0.735180i	$-0.262902\pi$
$601$	−178.947	−	309.945i	−0.297749	−	0.515716i	−0.975621	+	0.219464i	$0.929569\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	4.68975	−	2.70763i	0.00775164	−	0.00447541i
$606$		0			0
$607$	320.064	−	554.367i	0.527288	−	0.913290i	−0.472206	−	0.881488i	$-0.656542\pi$
$607$	320.064	−	554.367i	0.527288	−	0.913290i	0.999494	−	0.0318015i	$-0.0101245\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			16.7550i			0.0274223i
$612$		0			0
$613$	−246.093			−0.401457			−0.200729	−	0.979647i	$-0.564331\pi$
$613$	−246.093			−0.401457			−0.200729	+	0.979647i	$0.564331\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−309.912	−	178.928i	−0.502289	−	0.289997i	0.227369	−	0.973809i	$-0.426987\pi$
$617$	−309.912	−	178.928i	−0.502289	−	0.289997i	−0.729658	+	0.683812i	$0.760321\pi$
$618$		0			0
$619$	40.5053	+	70.1573i	0.0654368	+	0.113340i	0.896888	−	0.442258i	$-0.145823\pi$
$619$	40.5053	+	70.1573i	0.0654368	+	0.113340i	−0.831451	+	0.555598i	$0.812489\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−851.888	+	491.838i	−1.36740	+	0.789467i
$624$		0			0
$625$	−312.403	+	541.098i	−0.499845	+	0.865757i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	768.075i		−	1.22110i
$630$		0			0
$631$	−252.241			−0.399748			−0.199874	−	0.979822i	$-0.564053\pi$
$631$	−252.241			−0.399748			−0.199874	+	0.979822i	$0.564053\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−2.13175	−	1.23077i	−0.00335709	−	0.00193822i
$636$		0			0
$637$	−7.17147	−	12.4214i	−0.0112582	−	0.0194998i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	843.278	−	486.867i	1.31557	−	0.759542i	0.332554	−	0.943084i	$-0.392090\pi$
$641$	843.278	−	486.867i	1.31557	−	0.759542i	0.983012	+	0.183542i	$0.0587563\pi$
$642$		0			0
$643$	341.530	−	591.547i	0.531151	−	0.919980i	−0.468188	−	0.883629i	$-0.655093\pi$
$643$	341.530	−	591.547i	0.531151	−	0.919980i	0.999339	−	0.0363512i	$-0.0115735\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			390.640i			0.603771i	0.953344	+	0.301885i	$0.0976160\pi$
$647$			390.640i			0.603771i	−0.953344	+	0.301885i	$0.902384\pi$
$648$		0			0
$649$	−37.5081			−0.0577937
$650$		0			0
$651$		0			0
$652$		0			0
$653$	542.529	+	313.230i	0.830826	+	0.479678i	0.854135	−	0.520051i	$-0.174087\pi$
$653$	542.529	+	313.230i	0.830826	+	0.479678i	−0.0233093	+	0.999728i	$0.507420\pi$
$654$		0			0
$655$	−0.000805043	−	0.00139438i	−1.22907e−6	−	2.12882e-6i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	51.4518	−	29.7057i	0.0780755	−	0.0450769i	−0.460454	−	0.887684i	$-0.652313\pi$
$659$	51.4518	−	29.7057i	0.0780755	−	0.0450769i	0.538530	+	0.842607i	$0.318980\pi$
$660$		0			0
$661$	−425.950	+	737.767i	−0.644402	+	1.11614i	0.340037	+	0.940412i	$0.389560\pi$
$661$	−425.950	+	737.767i	−0.644402	+	1.11614i	−0.984439	+	0.175725i	$0.943773\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			15.7551i			0.0236918i
$666$		0			0
$667$	561.132			0.841277
$668$		0			0
$669$		0			0
$670$		0			0
$671$	240.728	+	138.985i	0.358760	+	0.207130i
$672$		0			0
$673$	210.489	+	364.577i	0.312762	+	0.541720i	0.978959	−	0.204056i	$-0.0654124\pi$
$673$	210.489	+	364.577i	0.312762	+	0.541720i	−0.666197	+	0.745776i	$0.732079\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	313.326	−	180.899i	0.462816	−	0.267207i	−0.250412	−	0.968139i	$-0.580566\pi$
$677$	313.326	−	180.899i	0.462816	−	0.267207i	0.713227	+	0.700933i	$0.247233\pi$
$678$		0			0
$679$	293.527	−	508.403i	0.432293	−	0.748753i
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	705.769i		−	1.03334i	−0.856186	−	0.516668i	$-0.827172\pi$
$683$		−	705.769i		−	1.03334i	0.856186	−	0.516668i	$-0.172828\pi$
$684$		0			0
$685$	0.0536179			7.82743e−5
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−18.9824	−	10.9595i	−0.0275506	−	0.0159063i
$690$		0			0
$691$	−345.384	−	598.222i	−0.499832	−	0.865734i	0.500168	−	0.865928i	$-0.333272\pi$
$691$	−345.384	−	598.222i	−0.499832	−	0.865734i	−1.00000		0.000194148i	$0.999938\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.99320	−	2.30547i	0.00574561	−	0.00331723i
$696$		0			0
$697$	737.703	−	1277.74i	1.05840	−	1.83320i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1213.04i		−	1.73045i	−0.501384	−	0.865225i	$-0.667176\pi$
$701$		−	1213.04i		−	1.73045i	0.501384	−	0.865225i	$-0.332824\pi$
$702$		0			0
$703$	1001.90			1.42518
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1526.22	−	881.161i	−2.15872	−	1.24634i
$708$		0			0
$709$	−226.667	−	392.598i	−0.319699	−	0.553735i	0.660726	−	0.750627i	$-0.270249\pi$
$709$	−226.667	−	392.598i	−0.319699	−	0.553735i	−0.980425	+	0.196892i	$0.936915\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1029.81	+	594.560i	−1.44433	+	0.833885i
$714$		0			0
$715$	0.0418908	−	0.0725570i	5.85885e−5	−	0.000101478i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	418.833i		−	0.582522i	−0.956644	−	0.291261i	$-0.905925\pi$
$719$		−	418.833i		−	0.582522i	0.956644	−	0.291261i	$-0.0940748\pi$
$720$		0			0
$721$	−225.421			−0.312650
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−350.753	−	202.508i	−0.483798	−	0.279321i
$726$		0			0
$727$	376.166	+	651.539i	0.517423	+	0.896203i	0.999795	+	0.0202365i	$0.00644191\pi$
$727$	376.166	+	651.539i	0.517423	+	0.896203i	−0.482372	+	0.875966i	$0.660225\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−179.284	+	103.509i	−0.245258	+	0.141600i
$732$		0			0
$733$	−255.863	+	443.168i	−0.349063	+	0.604595i	−0.986083	−	0.166252i	$-0.946833\pi$
$733$	−255.863	+	443.168i	−0.349063	+	0.604595i	0.637020	+	0.770847i	$0.280167\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		−	96.3737i		−	0.130765i
$738$		0			0
$739$	466.830			0.631705			0.315853	−	0.948808i	$-0.397710\pi$
$739$	466.830			0.631705			0.315853	+	0.948808i	$0.397710\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	546.320	+	315.418i	0.735290	+	0.424520i	0.820354	−	0.571856i	$-0.193776\pi$
$743$	546.320	+	315.418i	0.735290	+	0.424520i	−0.0850643	+	0.996375i	$0.527110\pi$
$744$		0			0
$745$	−0.443300	−	0.767818i	−0.000595034	−	0.00103063i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−182.249	+	105.222i	−0.243323	+	0.140483i
$750$		0			0
$751$	90.7172	−	157.127i	0.120795	−	0.209223i	−0.799286	−	0.600950i	$-0.794789\pi$
$751$	90.7172	−	157.127i	0.120795	−	0.209223i	0.920081	+	0.391727i	$0.128122\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$		−	4.15416i		−	0.00550220i
$756$		0			0
$757$	1381.82			1.82539			0.912696	−	0.408640i	$-0.133997\pi$
$757$	1381.82			1.82539			0.912696	+	0.408640i	$0.133997\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	590.093	+	340.690i	0.775418	+	0.447688i	0.834804	−	0.550548i	$-0.185581\pi$
$761$	590.093	+	340.690i	0.775418	+	0.447688i	−0.0593862	+	0.998235i	$0.518914\pi$
$762$		0			0
$763$	−711.471	−	1232.30i	−0.932466	−	1.61508i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	3.70791	−	2.14076i	0.00483430	−	0.00279109i
$768$		0			0
$769$	−45.1754	+	78.2461i	−0.0587457	+	0.101750i	−0.893903	−	0.448261i	$-0.852043\pi$
$769$	−45.1754	+	78.2461i	−0.0587457	+	0.101750i	0.835157	+	0.550012i	$0.185377\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	87.4025i		−	0.113069i	−0.998401	−	0.0565346i	$-0.981995\pi$
$773$		−	87.4025i		−	0.113069i	0.998401	−	0.0565346i	$-0.0180051\pi$
$774$		0			0
$775$	858.286			1.10747
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1666.72	+	962.284i	2.13957	+	1.23528i
$780$		0			0
$781$	−185.885	−	321.963i	−0.238010	−	0.412245i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.49274	−	4.90329i	0.0108188	−	0.00624622i
$786$		0			0
$787$	−366.731	+	635.196i	−0.465986	+	0.807111i	−0.999245	−	0.0388408i	$-0.987633\pi$
$787$	−366.731	+	635.196i	−0.465986	+	0.807111i	0.533260	+	0.845951i	$0.320967\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			340.602i			0.430597i
$792$		0			0
$793$	−31.7300			−0.0400126
$794$		0			0
$795$		0			0
$796$		0			0
$797$	853.500	+	492.768i	1.07089	+	0.618279i	0.928425	−	0.371521i	$-0.121164\pi$
$797$	853.500	+	492.768i	1.07089	+	0.618279i	0.142466	+	0.989800i	$0.454497\pi$
$798$		0			0
$799$	506.653	+	877.549i	0.634109	+	1.09831i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−253.617	+	146.426i	−0.315837	+	0.182349i
$804$		0			0
$805$	7.97200	−	13.8079i	0.00990310	−	0.0171527i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			459.662i			0.568185i	0.958797	+	0.284093i	$0.0916923\pi$
$809$			459.662i			0.568185i	−0.958797	+	0.284093i	$0.908308\pi$
$810$		0			0
$811$	−912.030			−1.12457			−0.562287	−	0.826942i	$-0.690078\pi$
$811$	−912.030			−1.12457			−0.562287	+	0.826942i	$0.690078\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.27951	−	4.20283i	−0.00893191	−	0.00515684i
$816$		0			0
$817$	−135.021	−	233.864i	−0.165265	−	0.286247i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	224.455	−	129.589i	0.273393	−	0.157843i	−0.357036	−	0.934091i	$-0.616213\pi$
$821$	224.455	−	129.589i	0.273393	−	0.157843i	0.630428	+	0.776247i	$0.282879\pi$
$822$		0			0
$823$	−721.885	+	1250.34i	−0.877139	+	1.51925i	−0.0226714	+	0.999743i	$0.507217\pi$
$823$	−721.885	+	1250.34i	−0.877139	+	1.51925i	−0.854467	+	0.519506i	$0.826116\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			1148.68i			1.38897i	0.719505	+	0.694487i	$0.244369\pi$
$827$			1148.68i			1.38897i	−0.719505	+	0.694487i	$0.755631\pi$
$828$		0			0
$829$	424.655			0.512250			0.256125	−	0.966644i	$-0.417554\pi$
$829$	424.655			0.512250			0.256125	+	0.966644i	$0.417554\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−751.216	−	433.715i	−0.901820	−	0.520666i
$834$		0			0
$835$	−6.32718	−	10.9590i	−0.00757746	−	0.0131245i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	52.2029	−	30.1394i	0.0622204	−	0.0359230i	−0.468567	−	0.883428i	$-0.655230\pi$
$839$	52.2029	−	30.1394i	0.0622204	−	0.0359230i	0.530787	+	0.847505i	$0.321896\pi$
$840$		0			0
$841$	−289.243	+	500.984i	−0.343928	+	0.595700i
$842$		0			0
$843$		0			0
$844$		0			0
$845$		−	8.57901i		−	0.0101527i
$846$		0			0
$847$	−965.291			−1.13966
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−878.078	−	506.959i	−1.03182	−	0.595721i
$852$		0			0
$853$	602.705	+	1043.92i	0.706571	+	1.22382i	0.966122	+	0.258087i	$0.0830923\pi$
$853$	602.705	+	1043.92i	0.706571	+	1.22382i	−0.259550	+	0.965730i	$0.583574\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−174.400	+	100.690i	−0.203500	+	0.117491i	−0.598287	−	0.801282i	$-0.704152\pi$
$857$	−174.400	+	100.690i	−0.203500	+	0.117491i	0.394787	+	0.918773i	$0.370818\pi$
$858$		0			0
$859$	−706.859	+	1224.31i	−0.822885	+	1.42528i	0.0806394	+	0.996743i	$0.474304\pi$
$859$	−706.859	+	1224.31i	−0.822885	+	1.42528i	−0.903525	+	0.428536i	$0.859030\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		−	299.383i		−	0.346909i	−0.984842	−	0.173455i	$-0.944507\pi$
$863$		−	299.383i		−	0.346909i	0.984842	−	0.173455i	$-0.0554930\pi$
$864$		0			0
$865$	6.89371			0.00796961
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−277.373	−	160.141i	−0.319186	−	0.184282i
$870$		0			0
$871$	5.50050	+	9.52714i	0.00631515	+	0.0109382i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−19.9336	+	11.5087i	−0.0227813	+	0.0131528i
$876$		0			0
$877$	−247.122	+	428.029i	−0.281782	+	0.488060i	−0.971824	−	0.235709i	$-0.924259\pi$
$877$	−247.122	+	428.029i	−0.281782	+	0.488060i	0.690042	+	0.723769i	$0.257592\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	211.533i		−	0.240105i	−0.992768	−	0.120053i	$-0.961694\pi$
$881$		−	211.533i		−	0.240105i	0.992768	−	0.120053i	$-0.0383063\pi$
$882$		0			0
$883$	410.167			0.464515			0.232258	−	0.972654i	$-0.425389\pi$
$883$	410.167			0.464515			0.232258	+	0.972654i	$0.425389\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	734.748	+	424.207i	0.828352	+	0.478249i	0.853288	−	0.521440i	$-0.174605\pi$
$887$	734.748	+	424.207i	0.828352	+	0.478249i	−0.0249363	+	0.999689i	$0.507938\pi$
$888$		0			0
$889$	219.390	+	379.994i	0.246782	+	0.427440i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1144.70	+	660.895i	−1.28186	+	0.740084i
$894$		0			0
$895$	−2.42545	+	4.20100i	−0.00271000	+	0.00469385i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			556.305i			0.618805i
$900$		0			0
$901$	−1325.61			−1.47126
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2.59337	+	1.49728i	0.00286560	+	0.00165445i
$906$		0			0
$907$	577.488	+	1000.24i	0.636701	+	1.10280i	0.986152	+	0.165844i	$0.0530347\pi$
$907$	577.488	+	1000.24i	0.636701	+	1.10280i	−0.349451	+	0.936955i	$0.613632\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−454.824	+	262.593i	−0.499258	+	0.288247i	−0.728407	−	0.685145i	$-0.759739\pi$
$911$	−454.824	+	262.593i	−0.499258	+	0.288247i	0.229149	+	0.973391i	$0.426406\pi$
$912$		0			0
$913$	89.2486	−	154.583i	0.0977531	−	0.169313i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			0.287005i			0.000312982i
$918$		0			0
$919$	−1548.07			−1.68452			−0.842260	−	0.539072i	$-0.818775\pi$
$919$	−1548.07			−1.68452			−0.842260	+	0.539072i	$0.818775\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	36.7519	+	21.2187i	0.0398179	+	0.0229889i
$924$		0			0
$925$	365.914	+	633.781i	0.395583	+	0.685169i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	531.821	−	307.047i	0.572466	−	0.330513i	−0.185668	−	0.982613i	$-0.559445\pi$
$929$	531.821	−	307.047i	0.572466	−	0.330513i	0.758134	+	0.652099i	$0.226111\pi$
$930$		0			0
$931$	565.752	−	979.911i	0.607682	−	1.05254i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	5.06692i		−	0.00541917i
$936$		0			0
$937$	−1763.19			−1.88174			−0.940868	−	0.338774i	$-0.889988\pi$
$937$	−1763.19			−1.88174			−0.940868	+	0.338774i	$0.889988\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	1201.52	+	693.697i	1.27685	+	0.737191i	0.976268	−	0.216564i	$-0.0694851\pi$
$941$	1201.52	+	693.697i	1.27685	+	0.737191i	0.300584	+	0.953755i	$0.402818\pi$
$942$		0			0
$943$	−973.824	−	1686.71i	−1.03269	−	1.78867i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	1073.21	−	619.615i	1.13327	−	0.654293i	0.188514	−	0.982071i	$-0.439633\pi$
$947$	1073.21	−	619.615i	1.13327	−	0.654293i	0.944755	+	0.327778i	$0.106300\pi$
$948$		0			0
$949$	16.7145	−	28.9503i	0.0176127	−	0.0305061i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	460.952i		−	0.483686i	−0.970315	−	0.241843i	$-0.922248\pi$
$953$		−	460.952i		−	0.483686i	0.970315	−	0.241843i	$-0.0777518\pi$
$954$		0			0
$955$	9.67383			0.0101297
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−8.27714	−	4.77881i	−0.00863101	−	0.00498312i
$960$		0			0
$961$	−108.946	−	188.700i	−0.113368	−	0.196358i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.466208	+	0.269166i	−0.000483117	+	0.000278928i
$966$		0			0
$967$	−9.76447	+	16.9126i	−0.0100977	+	0.0174897i	−0.871030	−	0.491230i	$-0.836548\pi$
$967$	−9.76447	+	16.9126i	−0.0100977	+	0.0174897i	0.860932	+	0.508719i	$0.169881\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$		−	509.199i		−	0.524406i	−0.965013	−	0.262203i	$-0.915551\pi$
$971$		−	509.199i		−	0.524406i	0.965013	−	0.262203i	$-0.0844491\pi$
$972$		0			0
$973$	−821.921			−0.844728
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−886.588	−	511.872i	−0.907460	−	0.523922i	−0.0278467	−	0.999612i	$-0.508865\pi$
$977$	−886.588	−	511.872i	−0.907460	−	0.523922i	−0.879613	+	0.475690i	$0.842198\pi$
$978$		0			0
$979$	206.333	+	357.379i	0.210759	+	0.365045i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1161.49	−	670.589i	1.18158	−	0.682187i	0.225202	−	0.974312i	$-0.427696\pi$
$983$	1161.49	−	670.589i	1.18158	−	0.682187i	0.956380	+	0.292126i	$0.0943625\pi$
$984$		0			0
$985$	5.48536	−	9.50092i	0.00556889	−	0.00964560i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			273.281i			0.276320i
$990$		0			0
$991$	505.274			0.509863			0.254931	−	0.966959i	$-0.417947\pi$
$991$	505.274			0.509863			0.254931	+	0.966959i	$0.417947\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	6.45507	+	3.72683i	0.00648750	+	0.00374556i
$996$		0			0
$997$	−878.908	−	1522.31i	−0.881553	−	1.52689i	−0.849615	−	0.527404i	$-0.823165\pi$
$997$	−878.908	−	1522.31i	−0.881553	−	1.52689i	−0.0319379	−	0.999490i	$-0.510168\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.3.q.j.449.2		8
3.2	odd	2		576.3.q.i.257.3		8
4.3	odd	2		1728.3.q.i.449.2		8
8.3	odd	2		432.3.q.e.17.3		8
8.5	even	2		216.3.m.b.17.3		8
9.2	odd	6	inner	1728.3.q.j.1601.2		8
9.7	even	3		576.3.q.i.65.3		8
12.11	even	2		576.3.q.j.257.2		8
24.5	odd	2		72.3.m.b.41.2	✓	8
24.11	even	2		144.3.q.e.113.3		8
36.7	odd	6		576.3.q.j.65.2		8
36.11	even	6		1728.3.q.i.1601.2		8
72.5	odd	6		648.3.e.c.161.5		8
72.11	even	6		432.3.q.e.305.3		8
72.13	even	6		648.3.e.c.161.4		8
72.29	odd	6		216.3.m.b.89.3		8
72.43	odd	6		144.3.q.e.65.3		8
72.59	even	6		1296.3.e.i.161.5		8
72.61	even	6		72.3.m.b.65.2	yes	8
72.67	odd	6		1296.3.e.i.161.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.3.m.b.41.2	✓	8	24.5	odd	2
72.3.m.b.65.2	yes	8	72.61	even	6
144.3.q.e.65.3		8	72.43	odd	6
144.3.q.e.113.3		8	24.11	even	2
216.3.m.b.17.3		8	8.5	even	2
216.3.m.b.89.3		8	72.29	odd	6
432.3.q.e.17.3		8	8.3	odd	2
432.3.q.e.305.3		8	72.11	even	6
576.3.q.i.65.3		8	9.7	even	3
576.3.q.i.257.3		8	3.2	odd	2
576.3.q.j.65.2		8	36.7	odd	6
576.3.q.j.257.2		8	12.11	even	2
648.3.e.c.161.4		8	72.13	even	6
648.3.e.c.161.5		8	72.5	odd	6
1296.3.e.i.161.4		8	72.67	odd	6
1296.3.e.i.161.5		8	72.59	even	6
1728.3.q.i.449.2		8	4.3	odd	2
1728.3.q.i.1601.2		8	36.11	even	6
1728.3.q.j.449.2		8	1.1	even	1	trivial
1728.3.q.j.1601.2		8	9.2	odd	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1728.q (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.0440114 - 0.0254100i) q^{5} +(4.52944 + 7.84521i) q^{7} +O(q^{10})\)	\(q+(-0.0440114 - 0.0254100i) q^{5} +(4.52944 + 7.84521i) q^{7} +(3.29117 - 1.90016i) q^{11} +(-0.216902 + 0.375686i) q^{13} +26.2355i q^{17} -34.2225 q^{19} +(29.9930 + 17.3164i) q^{23} +(-12.4987 - 21.6484i) q^{25} +(14.0316 - 8.10114i) q^{29} +(-17.1675 + 29.7350i) q^{31} -0.460372i q^{35} -29.2761 q^{37} +(-48.7026 - 28.1185i) q^{41} +(3.94539 + 6.83362i) q^{43} +(33.4489 - 19.3117i) q^{47} +(-16.5316 + 28.6335i) q^{49} +50.5273i q^{53} -0.193132 q^{55} +(-8.54743 - 4.93486i) q^{59} +(36.5718 + 63.3442i) q^{61} +(0.0190923 - 0.0110230i) q^{65} +(12.6797 - 21.9618i) q^{67} -97.8262i q^{71} -77.0599 q^{73} +(29.8143 + 17.2133i) q^{77} +(-42.1389 - 72.9868i) q^{79} +(40.6763 - 23.4845i) q^{83} +(0.666644 - 1.15466i) q^{85} +108.587i q^{89} -3.92978 q^{91} +(1.50618 + 0.869593i) q^{95} +(-32.4021 - 56.1221i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{5} + 6 q^{7}+O(q^{10}) \) 8 * q - 6 * q^5 + 6 * q^7	\( 8 q - 6 q^{5} + 6 q^{7} + 36 q^{11} - 14 q^{13} - 4 q^{19} + 102 q^{23} + 10 q^{25} - 114 q^{29} - 50 q^{31} - 120 q^{37} - 264 q^{41} + 28 q^{43} - 150 q^{47} + 94 q^{49} - 244 q^{55} - 108 q^{59} - 14 q^{61} + 198 q^{65} + 20 q^{67} - 76 q^{73} + 66 q^{77} + 26 q^{79} + 246 q^{83} + 224 q^{85} - 108 q^{91} + 456 q^{95} - 236 q^{97}+O(q^{100}) \) 8 * q - 6 * q^5 + 6 * q^7 + 36 * q^11 - 14 * q^13 - 4 * q^19 + 102 * q^23 + 10 * q^25 - 114 * q^29 - 50 * q^31 - 120 * q^37 - 264 * q^41 + 28 * q^43 - 150 * q^47 + 94 * q^49 - 244 * q^55 - 108 * q^59 - 14 * q^61 + 198 * q^65 + 20 * q^67 - 76 * q^73 + 66 * q^77 + 26 * q^79 + 246 * q^83 + 224 * q^85 - 108 * q^91 + 456 * q^95 - 236 * q^97

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